The distribution of Σ boundaries in sintered magnetite

Materials Science and Engineering, 100 (1988) 255-260

255

The Distribution of Z Boundaries in Sintered Magnetite SANGHO AHN and JEROME B. COHEN Department of Materials Science and Engineering and the Materials Research Center. Northwestern Universi~, Evanston. 1L 60201 (U.S.A.) (Received September 4, 1987)

Abstract

Grain boundaries in sintered magnetite were characterized with Kikuchi patterns from the neighboring grains, obtained by transmission electron microscopy. The misorientations were catalogued according to their X values, and their relative frequencies were compared with those for a random distribution (obtained by computer simulation). The experimentally determined distribution was essentially random; there was no dominance of low Z, boundaries. 1. Introduction

It is well established that a low-angle boundary can be described as a dislocation network, the Burgers' vectors of which are lattice translation vectors. As the misorientation angle between neighboring grains increases, the spacing of dislocations becomes smaller and the overlapping of dislocation cores is inevitable. Thereafter, the boundary structure cannot be described by a dislocation array. The coincidence site lattice (CSL) was proposed [1, 2] to describe such high-angle boundaries (in which the misorientation angles are more than about 15°). The CSL can be visualized as a mode II twin structure for a cubic crystal with rotation of 180 ° about some rational direction [hkt]. The CSL boundaries are usually referred to by their Z values, which is the inverse density of coincident sites. The most basic feature of a CSL description is that the boundary structure is periodic. Each of the low Z configurations corresponds to a local energy minimum, and the atoms adjust to preserve the CSL. Deviation from the CSL is accommodated by secondary dislocation networks in the same manner as the primary dislocation network accommodates the mismatch in a small-angle boundary [3]. The physical significance of the CSL can be presumed to be that the more atoms there are in the boundary that occupy the CSL, the better is the atomic fit, hence the lower is the boundary energy. This implies that 0025-5416/88/$3.50

low Z boundaries are energetically preferred to high Y boundaries. Therefore, if kinetically feasible, boundaries will eventually attain low Z values. There is still debate on the issue of the extent or even the validity of Z values for low energy boundaries [4]. Theoretical calculations of grain boundary energy using various interatomic potentials do not show any meaningful energy cusps at Y misorientations, thus the author argued the irrelevance of Z values to low energy boundaries [5]. Moreover, the calculation of grain boundary energy indicates that [001] twist boundaries in ceramic oxides with the NaC1 structure are barely stable owing to a repulsive force between the neighboring same kind of ions [6]. Restructured grain boundary models, containing systematic arrays of Schottky defects have been suggested [7], which stabilize the boundaries. However, the experimental observations of secondary dislocation networks near Z boundaries in metals [8] and ceramics [9] are as yet difficult to explain without assuming the direct relation of Y to a low energy configuration. There are also experiments performed with smoke particles [10] and sphere (or crystallite) rotation techniques [11, 12] which show that some of the Z boundaries are indeed preferred. In one crystallite rotation experiment [12], the trapping of boundaries on annealing gold was observed at the Z 5 misorientation. Also, secondary dislocation networks in gold bicrystals were observed at Z 5, Y 13 and Y 17 [8]. (The observation is believed to be limited by the resolution of the instrument, and thus the relative importance of specific Z values is still open to question.) This discrepancy in observations was attributed to the depth of the energy cusp. In the experiments described above, the crystals are free to rotate to attain a low energy configuration. However, in a sintered material, several factors determine the final boundary network, in addition to the restricted free rotation. Therefore, it is of interest to see if a preference for low Y boundaries occurs during sintering, and the present study © Elsevier Sequoia/Printed in The Netherlands

256 reports on results for sintered magnetite. For this purpose, the observed boundary distribution will be compared with a random distribution.

2. E x p e r i m e n t a l t e c h n i q u e s

2.1. Sample preparation Haematite powder (99.999% pure from Gallad Schlesinger, Carle Place, New York) was reduced to magnetite in an environment of Pco2/Pco = 33 at 973 K. The reduced powder was vibratory milled and then sieved to - 1 5 0 ktm. Samples were uniaxially pressed in a 12.5 mm double-action die at a pressure of 1.5 MPa. Sintering was performed at 1373 K for 1 h in the above flowing gas mixture. The density of the sintered samples was more than 90% of the theoretical value. The grain size of the samples was typically in the range of 5 - 1 5 / ~ m , as derived from measurements of average intercept lengths. Cylinders, 3 m m in diameter, were cut from the sintered samples with an ultrasonic cutter, and sliced into discs about 200 kLm thick with a diamond wheel. The discs were mechanically polished until the thickness was about 90/~m, followed by ion milling (at 5 kV and 3 m A with liquid nitrogen cooling) to produce specimens suitable for transmission electron microscopy on a Hitachi H - 7 0 0 H electron microscope. The milling rate was typically 2 - 3 / z m h - 1. The operating voltage of 200 kV was chosen to obtain good Kikuchi patterns from the relatively thick specimens. 2.2. Data acquisition More than 100 pairs of Kikuchi patterns were taken using a selected area diffraction (SAD) aperture. Each pair was taken from two neighboring grains without tilting the specimen. (Three intersecting Kikuchi lines must appear in one diffraction pattern to index the pattern unambiguously.) Once the patterns were indexed (and therefore the orientation of each grain was provided), the orientation relationship was determined following the method suggested by Young et al. [13]. These procedures (indexing and determining the orientation relationship) were carried out with the aid of a computer program [14] written for an IBM PC. The method involves a determination of the orientations of a reference frame in both crystals, which are the directions of the incident beam, and a marker in each crystal. The orientation relationship is then obtained by comparing these orientations in crystals 1 and 2. The major sources of error are

associated with establishing these reference frames. Typically, the beam axis can be ascertained to within 0.1 ° from a matrix analysis of Kikuchi patterns, and the marker direction to +0.05 °. These can result in an error of + 0.25 ° in the misorientation angle and _+0.1 ° in the misorientation axis [13]. Computer analysis with the aid of a digitizer to measure Kikuchi patterns can reduce the error in the beam axis to 0.05 ° [14J. In the present work, the same software as in ref. 14 was employed, but rather than rely on previous estimates of the error, the following procedure was adopted. The orientation of the reference frame was obtained several times, by repeating the analysis of a set of three Kikuchi pairs from two grains, and by choosing another set of three. Also included were sets taken at different times. The variation for repeated measurements of the same pair was less than 0.05 °, supporting the claims in ref. 14. However, the scatter from different sets was as much as _+0.5 °, and we chose this value as our error estimate (for both the misorientation axis and angle). The Z values were then catalogued by comparing the experimentally determined misorientations with those calculated and listed for Z ~<99 [15]. The deviation from the nearest CSL orientation for each boundary was determined using the relationship [16]

0-tan(tanA0

+ tan 2

--

(1)

2

where A 01 and A 02 define the deviation of the misorientation axis and angle respectively from those of the CSL. The maximum allowable deviation that can be accommodated by secondary dislocations was calculated following Brandon [1] A0 c = 15Z-1/2

(2)

where a misorientation angle of 15 ° was taken as the limit for a low-angle boundary (Z = 1). This angle was used to catalogue boundaries; i.e. within this angle, a boundary was considered a CSL.

2.3. Generating a random distribution of boundaries To detect any preference for certain Z boundaries, we calculated the percentage of such boundaries in a random distribution. A random distribution of boundaries is one that can be drawn from a random polycrystalline aggregate in which the orientation of each grain is equally probable. Alternatively, if any grain is chosen as a reference crystal, the orientation of a neighboring grain is ran-

257 dom, i.e. a random orientation relationship exists with respect to the reference crystal. The orientation relationship between grains can be represented by a rotational transformation, which is expressed mathematically by a 3 x 3 orthogonal matrix. Therefore random 3 × 3 orthogonal matrices were generated in a VAX 11-370 computer to represent randomly distributed grain boundaries. The construction of the matrix was based on the method suggested by Mackenzie and Thomson I I 7], which is to determine a set of three orthogonal unit vectors as the columns of the matrix. A unit vector X (x~, x,, x~) can be expressed in polar coordinates x I = sin 0 cos ¢ x, = sin 0 sin # X 3 = COS 0

where 0 ~< 0 ~< ~ and 0 4 ¢ ~<2 ~. Two random numbers between zero and unity were generated and multiplied by 3 and 27r to obtain random 0 and ¢ respectively. These determine a random unit vector as a first column of the matrix. Another random unit vector is obtained in the same manner, which forms a random plane together with the first random vector. Therefore, the second column vector is obtained from the cross-product of the two unit vectors after these were normalized. The third column is the cross-product of the first and the second column vectors. The rotation matrix is not unique because there are 24 symmetry operations associated with a cubic crystal; one rotation matrix generates 24 equivalent matrices describing the same orientation relationship. The one with the smallest value of rotation angle was taken to represent the orientation relationship of the bicrystal, and is referred to as a disorientation matrix. The rotation axis-angle were obtained from this disorientation matrix and compared with those of the CSL to determine the Z value. One hundred disorientation matrices were generated and catalogued. This population was chosen to be comparable with the number of grain boundaries observed in sintered magnetite samples, and this procedure was repeated 25 times to obtain an estimation of confidence (95%) for each occurrence of a particular X value.

3. Results Table 1 shows the distribution of the boundaries (2] ~<99) obtained from the simulation. This distribution was obtained for a range of A 0 c, to compare

with that in sintered samples, for which experimental error had to be taken into account. The distribution of Z boundaries among 106 boundaries in sintered samples is also given in Table 1. (The amount of addition to A 0~, implies an experimental error or a possible rotation of grains toward the configuration of c s g , and is discussed below.) As can be seen in Table 1, there is no significant difference between the experimental distribution and the simulation. The proportion of boundaries with Z ~<99 in sintered samples is 18.9')/,, compared with 17.5 _+ 1.5% for the simulation. When experimental errors of 0.5 ° and 1.0 ° are allowed in determining the distribution, these proportions are 26.4%, and 39.6",4, respectively for sintered samples, and 26.5 +. . .~. ~"~ J,,, and 36.8_+ 1.8,o,/ for the simulation. Also, the proportion of boundaries with Z ~<25 for both cases is about 1 3 % . Similar agreement can be found in Table 1 for Z > 99, for Z (27-99), or Z (3-25). The closeness of the agreement observed here indicates that the boundaries in the sintered samples are randomly distributed. Ahhough the probability of each occurrence of Z can have large variations because of the small sample size (and thus there might be Z values with mild cusps in energy), the comparison of these various Z ranges clearly shows that no Z values dominate. (In ref. 18, it is shown that for 400 simulated boundaries these ranges show statistical fluctuations of the order of only 2-3%.) To add support to the idea that the distribution of boundaries in the sintered samples was indeed random, the experimentally determined distribution of misorientation angles was also compared with those obtained from computer simulation and analytical calculations in ref. 19. This comparison is illustrated in Fig. 1, where the solid curve. + marks, and histogram represent the results of analytical calculation, simulation and the experimental observations respectively. It can be seen that the experimentally determined values are in good agreement with the other sets. Therefore, it can be concluded that the distribution of the boundaries in the present specimen is an essentially random distribution, without any strong preference for particular Y boundaries.

4. Discussion Although there is excellent agreement between the experimental observations and simulation, there is one discrepancy. The proportion of boundaries with Z ~<25. calculated analytically by Warrington [18], was about 11%, close to the value of 13% estimated in the present study. However, the propor-

258

TABLE l. magnetite

Comparison between the distribution of X boundaries from computer simulation and that observed in sintered

Z

Computer simulation Total no. of boundaries 100 × 25 Probability (%) _+95% confidence interval Allowed error (deg) 0.0

1 3 5 7 9 11 13 l5 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 Random Total proportion (%) of Z

6.3200 ± 0.8060 1.0800 ± 0.3637 1.3200 _+0.4722 0.2800 _+0.1833 0.7600 _+0.2653 0.4800 + 0.3081 0.3600 ± 0.1960 0.4400 __+0.2603 0.5600 _+0.2847 0.1600 ± 0.1497 0.3200 _+0.2509 0.2400 ± 0.1744 0.4800 ± 0.2344 0.1600 ± 0.1890 0.3600 -+ 0.2274 0.2400 -+ 0.2091 0.2800 ± 0.2166 0.1200 __+0.1327 0.0400 ± 0.0800 0.2800 + 0.2166 0.1200 ± 0.1327 0.0800__+ 0.1108 0.0400 ± 0.0800 0.1600 __+0.1497 0.2000 _+0.2000 0.2400 __+0.1744 0.1200 ± 0.1327 0.2000 ± 0.1633 0.2000 ± 0.2000 0.0400 ± 0.0800 0.0000 _+0.0000 0.2000 ± 0.2000 0.0800 _+0.1108 0.1200 ± 0.1327 0.1200 _+0.1327 0.2000 __+0.1633 0.1200 _+0.1327 0.2000 ± 0.1633 0.1600 ± 0.1497 0.0000 ± 0.0000 0.0400 _+0.0800 0.0000 ± 0.0000 0.2000 ± 0.2000 0.1200_+ 0.1327 0.0000 __+0.0000 0.0400 ± 0.0800 0.1600 _+0.1497 0.0800 ± 0.1108 0.0000 ± 0.0000 0.0000__+ 0.0000 82.4800 ± 1.5411 17.52 -+ 1.54

1 i0

0.5 6.1600 ± 0.8845 1.4400 + 0.4772 1.8800 ± 0.4806 0.2800 _+0.2166 0.7200_+ 0.3370 0.6800 ± 0.3208 0.8000 _+0.3055 0.4000 _+0.2828 0.6400 _+0.3441 0.2800 _+0.1833 0.3200 ± 0.1904 0.3600_+ 0.2800 0.4400 _+0.2603 0.6000__+ 0.2582 0.3200 -+ 0.2509 0.2800 ± 0.1833 0.4000 -+ 0.2582 0.1600 _+0.1497 0.2400 __0.2091 0.4400 ± 0.2332 0.4000 _+0.2000 0.1200 ± 0.I 327 0.3600 __+0.2274 0.2400 -+ 0.1744 0.4400 _+0.2026 0.4400__+ 0.2603 0.3600 ± 0.2274 0.6000 ± 0.2582 0.2000 -+ 0.1633 0.1600 __+0.1429 0.1200 _+0.1327 0.1600 + 0.1497 0.4000 _+0.2828 0.2400 ± 0.1744 0.4000 _+0.2828 0.6000 _+0.3651 0.3600 __+0.2551 0.3200 ± 0.2227 0.5600 _+0.3282 0.1600 ± 0.1890 0.5200 ± 0.2613 0.1200_+ 0.1327 0.3200 _+0.2509 0.2400 _+0.1744 0.2000 _+0.2000 0.2800 + 0.2166 0.2800 ± 0.1833 0.4800 ± 0.2857 0.2000 ± 0.1633 0.3200 -+ 0.1904 73.5600 -+ 1.9902 26.44 ± 1.99

t i o n o f l o w - a n g l e b o u n d a r i e s ( Y . = I ) is 2 . 2 8 % , w h i c h is c o n s i d e r a b l y s m a l l e r t h a n t h a t i n t h e c a s e o f t h e p r e s e n t s i m u l a t i o n (6.3 + 0.8%). (In ref. 18, t h e p r o p o r t i o n w a s 1 . 9 8 % o w i n g to a c h o i c e o f t h e limit for a low angle boundary of 1/4 radian instead

Sintered magnetite Total no. of boundaries 106 Number of occurrences Allowed error (deg)

6.8400 __+0.9979 1.7600 __+0.6955 2.4000 __+0.4899 0.6000 __+0.4320 0.7600__+0.3323 0.6800 _+0.2993 0.8400 _+0.2984 0.6800 _+0.3410 0.8800_+ 0.3124 0.4800 _+0.3290 0.7200 _+0.2948 0.1600_+0.2498 0.9600 _+0.3555 0.4000 _+0.2309 0.4800_+0.2613 0.5200 ± 0.3851 0.8400 ± 0.2984 0.6800 -+ 0.3208 0.4000 + 0.2582 1.0000 ± 0.4163 0.2000 ± 0.1633 0.3600 -+ 0.1960 0.4800 ± 0.2613 0.4000 ± 0.2309 0.6800 -+ 0.4430 0.6800 _+0.2993 0.2400 ± 0.1744 0.6400 __+0.3029 0.4000 ± 0.2582 0.5200 ± 0.2857 0.3600±0.2274 0.9200 ± 0.3250 0.5200 + 0.2344 0.4800 _+0.3290 0.6000±0.2828 0.4400 ± 0.2026 0.4000 _+0.2309 0.6800 _+0.3410 0.6000±0.3651 0.2800 __+0.2166 0.7200 _+0.3563 0.2400±0.2091 0.7200_+0.3745 0.2000±0.1633 0.0800 _+0.1108 0.6000 __+0.3055 0.6400±0.3630 0.6800 ± 0.3208 0.2000 ± 0.1633 0.7600 __+0.3323 63-2000 _+ 1.7889 36.80 _+ 1.79

0.0

0.5

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8 0 2 0 1 0 0 l 2 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 2 0 0 0 0 0 0 0 0 78

8 1 2 1 1 0 0 1 2 0 1 2 1 3 0 1 0 0 0 2 0 0 2 1 0 0 2 0 0 0 0 0 2 l 0 1 l l 0 1 1 2 0 0 0 0 0 l 0 0 64

18.9

26.4

39.6

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259

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Fig. 1. The density function for the angle of disorientation. T h e ordinate is the probability density. The solid curve, + marks and histogram represent the results of an analytical calculation [19], computer simulation and the observations with sintered magnetite respectively. The numbers at the top of the figure are the actual occurrence of the boundaries in each disorientation angle range in the sintercd sample.

2.5. Similar results (higher frequency of low-angle boundaries in simulation) were reported [20], but those researchers attributed it to a systematic error in their simulation, without further elaboration. Some 10 000 matrices were randomly generated in the present study to obtain the distribution of disorientation angles in Fig. 1. When smaller samples of 400 matrices were used, results showed the same distribution, but with higher fluctuations (Fig. 2). Thus the discrepancy does not seem to arise from small sample size; perhaps the analytical solution needs to be re-examined. The probability of Z boundaries was very sensitive to an increase in A 0~. For example, an allowance of 1.0 degrees of freedom toward a CSL configuration in both the misorientation axis and angle increases the total proportion of the Z boundaries by two. Furthermore, addition of the experimental error to A0 c would include boundaries which actually deviate more than the allowed A 0~, and could result in a higher fraction of Y boundaries in a random distribution (observed experimentally) than that from simulation. Therefore, the comparison between the experimental and computer results was made by examining the boundary distribution for a range of A 0~, but did not show any significant difference, and the total proportion of boundaries with Z ~<99 in the present specimen is estimated to be a fifth of all the boundaries. As discussed in the introduction, low Z boundaries may represent low-energy boundary structures. This would give rise to a high frequency of occur-

~,

30. DIS0RI~'TAT I ~

40.

511.

M.

I~.[

Fig. 2. Comparison between the computer simulation with 4/)0 matrices ( ) and that with 10 000 matrices c + ). (The mean and standard deviation of the distribution are 37.51 ° and 12.75 ° respectively for the case of 4011 matrices, compared with 37.49 ° and 12.63 ° for the case of 10001)

matrices.)

rence of such boundaries in a polycrystalline material if the grains are free to rotate. During the process of sintering, grain boundaries between particles exist from the onset of sintering. The porous specimen densities until sufficiently large interparticle contacts are formed so that boundaries can move (grain growth), driven by the excess surface energy of the boundary. Particle (or grain) rotation (especially in the early stages of sintering) has been detected [21 ]. Considering this possibility for grain rearrangement, slight orientation changes (less than about 1.0 °) of the grains to lower the boundary energy would result in a significant increase in the proportion of Z boundaries, assuming that subsequent grain growth would maintain the initial proportion. Therefore, the absence of any large preference for Z boundaries indicates that the energy difference between these and a random array of boundaries is not large enough to alter the boundary structure markedly. There have been no experimental reports of any high proportion of Z boundaries except Z 3 related boundaries (such as Z9, Z27 etc.) [22], which are the result of annealing twins produced abundantly during strain annealing of face-centered cubic (f.c.c.) materials. In fact, there have not been many reports about the distribution of Z boundaries from a sufficiently large number of grains to be statistically reliable. Nevertheless, results [23, 24] do show proportions of Z boundaries similar to those expected from a random distribution. It should also be noted that the absence of texture in a polycrystal does not always imply a ran-

26(} d o m distribution of the orientations of grains or grain boundaries. For example, m o r e than half of the b o u n d a r i e s in strain-annealed polycrystalline nickel were r e p o r t e d to be Y 3 related b o u n d a r i e s in spite of the lack of texture [22]. T h e r a n d o m n e s s of the orientations of grains in a polycrystal can be verified only by investigating the orientation relationships of the grains. T h e results of the present study prove that sintered magnetite is an essentially r a n d o m polycrystal aggregate.

Acknowledgments This research was s p o n s o r e d by the Materials R e s e a r c h C e n t e r at N o r t h w e s t e r n University, supp o r t e d under the N S F - M R C p r o g r a m , through grant D M R 8216972. T h e authors would like to thank Q u e n t i n R o b i n s o n of N o r t h w e s t e r n University for providing sintered samples.

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4 R J. Goodhew, in R. W. Balluffi (ed.), Grain Boundary Structure and Kinetics, American Society for Metals, Metals Park, OH, 1980, p. 155. 5 D. Wolf, Acta Metall., 32(1984) 245. 6 D. Wolf, J. Am. Ceram. Soc., 67(1984) 1. 7 D. W. Tasker and D. M. Dully, Philos. Mag. A, 47(1983) L45. 8 T. Schober and R. W, Balluffi, Philos. Mag. A, 21 (1970) 109. 9 C. P. Sun and R. W. Balluffi, Philos. Mag. A, 46 (1982) 49. 10 P. Chaudhari and J. W. Mattewes, J. Appl. Phys., 42 ( 1971 ) 3063. 11 G. Herrmann, H. Gleiter and G. Baro, Acre Metall., 24 (1976) 353. 12 S. Chan and R. W. Balluffi, Acta Metall., 33(1985) 1113. 13 C. T. Young, J. H. Steele, Jr. and J. L. Lytton, Metall. Trans., 4(1973) 2081. 14 P. Heilmann, W. A. T. Clark and D. A. Rigney, Ultramicroscopy, 9(1982) 365. 15 H. Mykura, in R. W. Balluffi (ed.), Grain Boundary Structure and Kinetics, American Society for Metals, Metals Park, OH, 1980, p. 445. 16 H. Kokawa, T. Watanabe and S. Karashima, Philos. Mag. A, 44(1981) 1239. 17 J. K. Mackenzie and M. J. Thomson, Biometrika, 44 (1957) 205. 18 D. H. Warrington, J. Micros& 102(1974) 301. 19 J. K. Mackenzie, Biometrika, 45 (1958) 229. 20 D. H. Warrington and M. Boon, Acre Metall., 23 (1975) 599. 21 H.E. Exner, Rev. PowderMet. Phys. Ceram., 1(1979) 1. 22 L.C. Lira and R. Raj, Acta Metall., 32 (1984) 1171. 23 T. Watanabe, Metall. Trans. A, 14(1983) 531. 24 D.J. Dingley and R. C, Pond, Acta Metall., 27(1979) 667.